POP Steering parameterization

Eddy diffusivity implementation by Bates, Tulloch, Marshall, and Ferrari

Steering Experiments 027 (Vertical integration changed from 100m-2000m to 100m-1000m

Sensitivity experiment where the vertical limit of integration for the Eady time scale was changed to 100m-1000m. In control exp 018 this limit was 100m to 2000m. This experiment does not include the fix to the eddy phase speed incorporated into exp 026

POP standard diagnostics for Exp 027

Time averaged diagnostics for first year (249) Exp 027

Exp 027 vs giaf control

\(\color{green}{\quad CESM \quad Implementation \quad of \quad Steering \quad Parameterization}\)

During the development of the parameterization several tuning modifications were made to the terms listed above to better match the observations.

  • New implementation of Eady Growth rate (sigma) that avoids undefined values at the equator. See alternate derivation at end of this page.

\(\quad\quad\quad \sigma_{vi} = {f \over \sqrt{R_i}}\) is replace by \(\sigma_{vi} = \quad {{\require\cancel \cancel f m^2} \over \cancel f N}\)

  • Don't use Rhine Scale in calculation of \(L_{eddy}\)

\(\quad\quad\quad L_{eddy} = min (L_r, L_{req}, \cancel{L_{Rh}})\)

  • Eddy phase speed calculation modified to avoid undefined value at the equator and limited to better match Bates paper.

\(\quad\quad\quad \cancel{c = - \beta * {L_r^2}}\) is replace with \(c = max(- \beta * L_{eddy}^2,-20)\)

  • \(u_{rms}\) is hard to capture in this implememtation due to the coarse model resolution and use of mean vs resolved states. We imposed a \(5 cm/sec\) minimum on \(U_{rms}\) to tune to surface obs of Bates and included a scaling constant \(\alpha\) to handle the bias introduced by using mean state instead of resolved state.

\(\quad\quad\quad u_{rms} = max(5.,alpha*\sigma_{vi}*L_r)\)

\(\quad\quad\quad \alpha=4\) The scaling constant for \(U_{rms}\)

  • The Eddy diffusivity calculated by our implementation was weak so we bumped up \(\Gamma\) from .35 to 1.75 (5x increase)

\(\quad\quad\quad \Gamma=.35\) replaced with \(\Gamma = 1.75\)

\(\color{green} {Parameterizing \quad the\quad Eddy\quad Length\quad Scale} \)

NOTES:

  • \(K=u_{rms}∗{\Gamma * \color{red}{L_{eddy}} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)}\)

  • \(L_{eddy} = min (L_r, L_{req}, \require{cancel}\cancel{L_{Rh}})\)

  • Since the two length scales we are using in the parameterization of \(L_{eddy}\) depend on the Baroclinic wave speed \(c_r\) we will first check \(c_r\) against Chelton 1998.

\(\quad\quad\quad\quad\quad\quad\quad\) First Baroclinic Wave Speed \(c_r\) CESM

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\(\quad\quad\quad\quad\quad\quad\) Chelton Baroclinic gravity wave phase speed

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\(\quad\quad\quad\) The barclinic wave speed looks reasonable so lets compare the Paramaterized Rossby Radius to Chelton

\(\quad\quad\quad\quad\quad\quad\) First Baroclinic Rossby Radius CESM \(\quad (L_{eddy})\)

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\(\color{green} {Parameterizing \quad Zonal\quad Eddy\quad Phase\quad Speed}\quad (\color{red}{c})\)

NOTES:

  • \(K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - \color{red}{c}|^2 /u_{rms}^2 (z=0)}\)

  • \(\require{cancel}\cancel{c = - \beta * {L_r^2}}\) \(L_r\) too high at equator

  • \(c = - \beta * L_{eddy}^2\)

\(\quad\quad\quad\quad\quad\) Eddy Phase speed CESM

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\(\quad \quad \quad \quad \)Compared to Hughes, the phase speed is much too high near the equator.

\(\quad \quad \quad \quad \)The first tuning mod is to limit the phase speed to 20cm/s.

\(\quad \quad \quad \quad \)Eddy Phase speed CESM \((c = max(- \beta * L_{eddy}^2,-20) )\)

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\(\quad\quad\quad\quad\quad\quad\) Hughes Phase Speed (cm/s) from Tulloch Marshall Smith '09

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\(\color{green}{Zonal\quad Velocity \quad Term}\quad (\color{red}{u_{mean}})\)

NOTES:

  • \(K=u_{rms}∗{\Gamma * L_{eddy} \over (1 + b1 * |\color{red}{u_{mean}} - c|^2 /u_{rms}^2 (z=0))}\)

  • Here \(U_{mean}\) is just CESM surface velocity.

\(\quad\quad \quad \quad\quad \) Zonal Velocity CESM

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\(\color{green}{(U-c)\quad Term}\)

NOTES:

  • \((u-c)\) limited to 20cm/s

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\) (U-c) Bates et al

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\(\color{green}{(U-c)^2 \quad Term}\)

NOTES:

  • \(c = max(- \beta * L_{eddy}^2,-20)\)

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad (U-c)^2\) Bates et al

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\(\color{green}{Parameterizing \quad Eady \quad Growth \quad Rate}\quad \color{red}{\sigma} \quad\quad\)

NOTES:

  • Now integrated of 100m-1000m (previously 100m-2000m)

  • The original derivation of \(\sigma_{vi}\) goes to 0 at the equator because of \(f\)

\(\quad\quad\quad\quad\quad\sigma_{vi} = {f \over \sqrt{R_i}}\)

  • The new derivation of \(\sigma_{vi}\):

\(\quad\quad\quad\quad\quad R_i = {f^2 N^2 \over {\underbrace{{ {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial y})^2 + {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial z})^2} }_\text{m^4}}}\quad\quad = \quad\quad {f^2N^2 \over m^4} \)

\(\quad\quad\quad\quad\quad \sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)

\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) CESM

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\(\quad\quad\quad\quad\quad \)Eady Growth Rate \((\sigma) \quad\) Bates et al

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\(\color{green}{Parameterizing \quad RMS \quad eddy \quad velocity}\quad \color{red}{u_{rms}}\)

NOTES:

  • \(K=\color{red}{u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2(z=0)}\)

  • \(u_{rms} = alpha*{\sigma_{vi}}*L_{eddy}\)

  • alpha (scaling constant) = 4

  • \(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)

  • \(u_{rms}\) is limited to 5 cm/s $ max(u_{rms},5.)$

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity \((u_{rms})\quad\) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms} \) Bates et al

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\(\color{green}{u_{rms}^2 \quad Term} \quad\)

NOTES:

  • \(K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /\color{red}{u_{rms}^2} (z=0)} \)
  • \(u_{rms}^2\) is a surface value

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Velocity Squared \((u_{rms}^2)\quad\) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad u_{rms}^2 \) Bates et al

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\(\color{green}{{(U-c)}^2 \over {u_{rms}^2}} \quad \color{green}{Term}\)

NOTES:

  • \(K={u_{rms}}∗{\Gamma * L_{eddy} \over (1 + b1 * \color{red}{|u_{mean} - c|^2/{u_{rms}^2} (z=0)}} \)
  • \(c = max(- \beta * L_{eddy}^2,-20)\)
  • \(u_{rms} = max(u_{rms},5.)\)
  • \(u_{rms}^2\) is a surface value

\(\quad\quad\quad\quad\quad\quad\) \({(U-c)}^2 \over {u_{rms}^2} \quad\) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad {|u-c|^2 \over u_{rms}^2} \) Bates et al

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\(\color{green}{\quad Suppression \quad factor = {1 \over (1 + b1 * |\bar u - c|^2 /u_{rms (z=0)}^2 )}}\)

NOTES:

  • \(c = max(- \beta * L_{eddy}^2,-20)\)
  • \(u_{rms} = max(u_{rms},5.)\)
  • \(u_{rms}^2\) is a surface value
  • \(b1\) (scaling constant) = 4.

\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor CESM

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\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor Bates

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\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\) Suppression Factor (Zonal x depth) Bates

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\(\color{green} {\quad Parameterizing \quad The \quad Mixing \quad Term \quad }\color{red}{L_{mix}}\)

NOTES:

  • \(\color{red}{L_{mix}} = \Gamma * L_{eddy} * Suppression\)

  • \(Suppression= {1 \over (1 + b1 * |u_{mean} - c|^2 /u_{rms (z=0)}^2 )}\)

  • \(\color{red}{L_{mix}} = {\Gamma * L_{eddy} \over (1 + b1 * |u_{mean} - c|^2 /u_{rms}^2 (z=0)}\)

  • \(\Gamma = 1.75\) (Tuning mod: the original Gamma of .35 produced a Kappa with the correct structure but too weak.

\(\quad\quad\quad\quad\quad\quad\quad\) LMIX CESM

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\(\color{green}{\quad Parameterizing \quad Eddy \quad Diffusivity \quad} (\color{red}{K})\)

NOTES:

  • \(\color{red}{K}=u_{rms}∗L_{mix}\)

\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) CESM

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\(\quad\quad\quad\quad\quad\quad\quad\) Eddy Diffusivity (K) Bates

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\(\quad\quad\quad\quad\quad\quad\quad\) (Zonal x Depth) CESM

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\) K limited to (100 < K < 10000)

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Here is the Bate's version

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\(\quad\quad\quad\quad\quad\quad\quad\) Zonal average of the N2 normalized scaling CESM

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The alternate derivation of \(\sigma_{vi}\):

\(R_i = {N^2\over{(\frac{\partial u}{\partial z})^2+(\frac{\partial v}{\partial z})^2}}\)

\(N^2={-g \over \rho_0 }\frac{\partial \rho}{\partial z}\)

After hydrostatic and geostrophic approximations

\(f \frac {\partial v}{\partial z} = {{-g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad f \frac {\partial u}{\partial z} = {{g \over \rho_0 }\frac {\partial \rho}{\partial y}} \)

so

\(\frac {\partial v}{\partial z} = {{-1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial x}}; \quad\quad \frac {\partial u}{\partial z} = {{1\over f}{g \over \rho_0 }\frac {\partial \rho}{\partial y}}\)

\(R_i = {f^2 N^2 \over {\underbrace{{ {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial y})^2 + {g^2 \over \rho_0^2 } ( \frac{\partial \rho}{\partial z})^2} }_\text{m^4}}}\quad\quad = \quad\quad {f^2N^2 \over m^4} \)

\(\sigma_{vi} = {f \over \sqrt{R_i}}\quad = \quad {f \over \sqrt{{f^2N^2 \over m^4}}}\quad = \quad {{\cancel f m^2} \over \cancel f N}\)

\(RX_1 = RX_{east} = \Delta\rho_x = \rho_{i+1,j} - \rho_{i,j}\)

\(RY_1 = RY_{north} = \Delta\rho_y = \rho_{i,j+1} - \rho_{i,j}\)

\(RZ_1 = RZ_{k+1} = \Delta\rho_z = \rho_{k} - \rho_{k+1}\)

\(\displaystyle{1 \over L_{R_i}} \displaystyle\int_{1000m}^{100m} \left\lbrace { {-g\over\rho_0}{\frac {\partial \rho} {\partial z}} \over { {g^2 \over \rho_0^2 } \left[( \frac{\partial \rho}{\partial y})^2 + ( \frac{\partial \rho}{\partial z})^2\right] } } \right\rbrace dz\)

Note: missing \(f^2\) which will be cancelled when forming \(\sigma_{vi}\)

\(\quad\quad\) so \(\cdots\) this is not \(R_i\)

Implementation notes

Numerator : Top \(= -grav * RZ_{SAVE}(\cdots k+1) * dzwr(k)\)

Denominator :

\(\begin{align} work1 = p25 & * ( RX(..,i_{east},k)^2 \\ & + RX(..,i_{west},k)^2 \\ & + RX(..,i_{east},k+1)^2 \\ & + RX(..,i_{west},k+1)^2 ) / DXT(i,j)^2 \\ \end{align}\)

\(\begin{align} work2 = p25 & * ( RY(..,j_{north},k)^2 \\ & + RY(..,j_{south},k)^2 \\ & + RY(..,j_{north},k+1)^2 \\ & + RY(..,j_{south},k+1)^2 / DYT(i,j)^2 \\ \end{align}\)

\(\begin{align} work3 = {\left( TOP \over (grav^2*(work1+work2))\right)}*dzw(k) \end{align}\)

Notes:

1)Need to be careful at top and bottom of ocean
2)Accurate dzw(k) for each (i,j) to form $L_{R_i}$
3) When constructing $sigma$ itself, use $RZ_{SAVE}$ with a minimum N value
4) use eps2

Posted by John Truesdale